Thread by @FreyaHolmer

the axis-angle representation of 3D rotation

a single 3D vector, where
• its direction v̂ is the axis of rotation
• its magnitude ||v|| is the angle to rotate by

this can represent any 3D rotation, but you'll most commonly see it represent angular velocity, where the magnitude is the angular speed

this is because unlike quaternions, axis-angle can go beyond a full turn, just like a number representing a single angle can

when representing angular velocity this way, you can also easily add angular velocities together through simple vector addition, decompose it, and project it!

which is pretty neat if you ask me

I've personally found the axis-angle representation incredibly useful when dealing with 3D rotations, and you can easily convert them to quaternions, and in turn use those, without really knowing how quaternions work

I talk about it at length here

https://twitter.com/FreyaHolmer/status/1443274220604166147

many people ask me if you have to learn quaternions to deal with 3D rotation

if you're using an existing game engine, then no, definitely not, which is good news bc quaternions are pretty complex

(no pun intended)

https://twitter.com/FreyaHolmer/status/1443274224005746696

✨ bonus tweet to pre-emptively avoid nitpickers ✨

axis-angle is the general concept of storing an axis and an angle to represent a rotation

encoding them as a single 3D vector is a special case, usually called a rotation vector or euler vector

("not all axis-angle" etc.)

✨ bonus tweet to pre-emptively avoid nitpickers ✨

axis-angle and quaternions can both represent any 3D orientation, but they have tradeoffs that has made most engines prefer quaternions for orientation, only using axis-angle when in need of rotations beyond a full turn

✨ bonus tweet to pre-emptively avoid nitpickers ✨

this rotation vector is not actually a true vector, but a pseudovector, due to its behavior in certain circumstances

eg. the angular velocity of a rolling object remains the same, even if you mirror the object along that axis

✨ bonus tweet to reactively avoid even more nitpickers ✨

yes yes pseudovectors are better represented using bivectors in geometric algebra which is on its own also pretty complicated and something something quaternions are more intuitively represented w. GA & rotors etc. etc.

imagine having to recursively explain every single concept adjacent to a concept in order to explain a concept

see this is the trap many educators fall into, and then every educational resource eventually becomes something with encyclopedic generalized coverage that is so dense people will struggle to even start understanding it

I will not falter

I will stay on my course

I will suffer through the consequences of making well-designed educational resources for my actual target audience

to make it for those actually looking to learn the concept, instead of targeting the audience that knows it already